Question

Figure, shows a sector of a circle, centre O, containing an angle ${\theta }^o$. Prove that Perimeter of the shaded region is $r(\tan \theta +\sec \theta +\frac{\pi \theta }{180}-1)$.

Answer 1

Length of the arc of the circle = $\frac{\pi \theta }{180}....i$

Length of the line which is also tangent to the circle = $r\tan \theta .....ii$

Length of the line which passes through the centre = $r\sec \theta ......iii$

Perimeter is $i+ii+iii-r$ as the line which passes through the centre has an additional radius value to it.

$\frac{\pi \theta }{180}+r\tan \theta +r\sec \theta -r$